Ackermann Function
An Ackermann Function is a total computable function that is not a primitive recursive function.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Ackermann_function Retrieved:2014-4-21.
- In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive.
After Ackermann's publication of his function (which had three nonnegative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function, is defined as follows for nonnegative integers m and n :[math]\displaystyle{ A(m, n) = \begin{cases} n+1 & \mbox{if } m = 0 \\ A(m-1, 1) & \mbox{if } m \gt 0 \mbox{ and } n = 0 \\ A(m-1, A(m, n-1)) & \mbox{if } m \gt 0 \mbox{ and } n \gt 0. \end{cases} }[/math] Its value grows rapidly, even for small inputs. For example A(4,2) is an integer of 19,729 decimal digits. [1]
- In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive.